(i) The vector equation of the line passing through A and B is \(\mathbf{r} = 2\mathbf{i} + \mathbf{j} + 3\mathbf{k} + \lambda(2\mathbf{i} - 2\mathbf{k})\).

Equating the components of the line l and the line through A and B gives:

\(4 + \mu = 2 + 2\lambda\)

\(6 + 2\mu = 1\)

\(-2\mu = 3 - 3\lambda\)

Solving these equations, we find that they are inconsistent, confirming that the lines do not intersect.

(ii) The direction vector for \(\overrightarrow{AP}\) is \((2 + t, 5 + 2t, -3 - 2t)\).

Using the cosine of the angle formula, \(\cos 120^{\circ} = -\frac{1}{2}\), we set up the equation:

\(\frac{(2 + t)(2) + (5 + 2t)(0) + (-3 - 2t)(-2)}{\sqrt{(2 + t)^2 + (5 + 2t)^2 + (-3 - 2t)^2} \cdot \sqrt{4 + 0 + 4}} = -\frac{1}{2}\)

Simplifying, we obtain \(3t^2 + 8t + 4 = 0\).

Solving this quadratic equation, we find \(t = -2\) or \(t = -\frac{2}{3}\). Using \(t = -2\), the position vector of P is \(2\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\).